Classes of finite groups with generalized subnormal cyclic primary subgroups

We study the properties of the classes $v_\pi\mathfrak H(v^*_\pi\mathfrak H)$ of finite groups whose all cyclic primary $\pi$-subgroups are $\mathfrak H$-subnormal (respectively, $\mathrm K$-$\mathfrak H$-subnormal) for a set of primes $\pi$ and a hereditary homomorph $\mathfrak H$. It is established that $v_\pi\mathfrak F$ is a hereditary saturated formation if $\mathfrak F$ is a hereditary saturated formation. We in particular obtain some new criteria for the $p$-nilpotency and $\phi$-dispersivity of finite groups. A characterization of formations with Shemetkov property is obtained in the class of all finite soluble groups.